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A high-order finite volume method with improved isotherms reconstruction for the computation of multiphase flows using the Navier–Stokes–Korteweg equations

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Science Arts & Métiers (SAM)

is an open access repository that collects the work of Arts et Métiers Institute of Technology researchers and makes it freely available over the web where possible.

This is an author-deposited version published in: https://sam.ensam.eu

Handle ID: .http://hdl.handle.net/10985/18007

To cite this version :

Abel MARTÍNEZ, Luis RAMÍREZ, Xesús NOGUEIRA, Sofiane KHELLADI, Fermín NAVARRINA -A high-order finite volume method with improved isotherms reconstruction for the computation of multiphase flows using the Navier–Stokes–Korteweg equations - Computers & Mathematics with Applications - Vol. 79, n°3, p.673-696 - 2020

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A high-order finite volume method with improved

isotherms reconstruction for the computation of

multiphase flows using the Navier-Stokes-Korteweg

equations

Abel Mart´ıneza, Luis Ram´ıreza, Xes´us Nogueiraa,∗, Sofiane Khelladib, Ferm´ın Navarrinaa

aGroup of Numerical Methods in Engineering, Universidade da Coru˜na, Campus de

Elvi˜na, 15071, A Coru˜na, Spain

bLaboratoire de Dynamique des Fluides, Arts et M´etiers ParisTech, 151 Boulevard de

l’Hˆopital. 75013 Paris, France.

Abstract

In this work we solve the Navier-Stokes-Korteweg (NSK) equations to simulate a two-phase fluid with phase change. We use these equations on a

diffuse interface approach, where the properties of the fluid vary continuously across the interface that separates the different phases. The model is able to

describe the behavior of both phases with the same set of equations, and it is also able to handle problems with great changes in the topology of the

prob-lem. However, high-order derivatives are present in NSK equations, which is a difficulty for the design of a numerical method to solve the problem. Here,

we propose the use of a high-order Finite Volume method with Moving Least Squares approximations to handle high-order derivatives and solve the NSK

equations. Moreover, a new methodology to obtain accurate equations of state is presented. In this method, we use any accurate equation of state for

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the pure phases. Under the saturation curve, a B-spline reconstruction

ful-filling a given set of thermodynamic criteria is performed. The new EOS can be used for computations using diffuse interface modeling. Several numerical

examples to show the accuracy of the new approach are presented.

Keywords: Diffuse interface, High-order methods, Finite Volume, Moving Least Squares, Phase transition

1. Introduction

Phase change and coexistence between two phases of the same substance

in the same space are phenomena which often takes place in engineering. Multiphase flow simulation is currently an area of extensive research, in which

great progress has been made recently.

Sharp-interface and diffuse-interface models are the two mayor branches

in multiphase flow modeling. The first ones [1, 2, 3] simulate different phases of the fluid as separated, therefore in the contact region between phases

(known as the interface) a discontinuity in the properties of the fluid is pro-duced. The interaction between different phases is achieved by imposing

balance equations as boundary conditions in the interface, which has to be solved at the same time that the system of partial differential equations. As

the interface is able to displace, the numerical simulation needs algorithms to track its position each time step.

Diffuse-interface models, or phase-field models [4, 5, 6, 7], represent a different approach. These models establish that in the interface there is

a smooth and continuous transition between the properties of each phase, and therefore it has non-zero thickness. The first advantage is that diffuse

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interface modeling has solid foundations in thermodynamics and statistical

mechanics. Moreover, in these methods the interface is part of the solution of the governing equations, thus the same system of equations can be applied

to the entire domain, avoiding expensive algorithms for tracking the interface and being capable of reproducing numerically problems with large topological

changes. The main problem of diffuse-interface models is that the governing equations are highly non-lineal, and to be realistic the interface must be

extremely thin, so they require very fine meshes or adaptive mesh techniques. One mathematical model to simulate multiphase flow is the

Navier-Stokes-Korteweg model (NSK), which is a highly non-lineal system belonging to the diffuse interface models type. One drawback of this model is the third-order

partial-differential equations that appear in the formulation [8], difficult to implement numerically.

In literature we find some numerical solutions of the Navier-Stokes Ko-rteweg equations. The Finite Difference method (FD) has been successfully

used to solve the NSK system [9]. However, it has limitations for its use in complex geometries, which are commonly found in engineering. To

over-come this limitation, other approaches have been proposed to compute the NSK system using the Discontinuous Galerkin method [10, 11, 12, 13] or

Isogeometric analysis [14]. In this work, we propose a high-order Finite Vol-ume method based on Moving Least Square approximations (FV-MLS) for

the resolution of the NSK system on unstructured grids [15, 16, 17, 18, 19]. Moving Least Squares (MLS) [20] approximations are used for the calculation

of the gradients and successive derivatives required for the reconstruction of the variables inside each control volume, an interpolation technique widely

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used in meshfree methods [21, 22, 23].

In diffuse interface modeling, a single equation of state (EOS) is required to describe the thermodynamic properties of the different phases that appear

in the problem. The most commonly used equation which reproduce phase change between two different states of matter is the van der Waals equation

of state [24, 25]. Although this equation of state is accurate enough for a great number of substances, it is very inaccurate for the case of water. In

this work, we also propose a new algorithm to obtain new accurate equations of state suitable for diffuse interface modeling. This is achieved by using

more accurate equations to represent the different phases of the substance and creating new expressions to ensure a continuous transition between the

pure phases.

The outline of this paper is as follows: First of all, the numerical method

is presented. Then, we show a new method to obtain accurate equations of states. Finally, a few numerical results are presented in Section 4, and then,

conclusions are drawn.

2. Numerical method

2.1. Navier-Stokes-Korteweg equations

The isothermal Navier-Stokes Korteweg (NSK) equations, can be written

in non-dimensional form [26] as system of conservation laws as

∂uuu

∂t + ∇∇∇ · FFF H

HH− FFFEEE− FFFKKK = 000

where uuu is the vector of variables, FFFHHH is the inviscid flux vector, FFFEEE is the viscous flux vector and FFFKKK is the Korteweg flux vector.

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u uu =          b ρ b ρvbx b ρvby          F FFHHH x =      b ρvbx b ρvbx2+ bP b ρvbxvby      , FFFHHH y =      b ρvby b ρvbxvby b ρvby2+ bP      FFFEEEx = 1 Re      0 2∂xvbx− 2 3(∂xvbx+ ∂yvby) (∂yvbx+ ∂xvby)      , FFFEEEy = 1 Re      0 (∂yvbx+ ∂xvby) 2∂yvby− 2 3µ (∂b xvbx+ ∂yvby)      FFFKKKx = 1 W e      0 b ρ∆ρ +b 12|∇ρ|b2− ∂xρ∂b xρb  − (∂xρ∂b yρ)b      , FFFKKKy = 1 W e      0 − (∂yρ∂b xρ)b ρ∆ρ +b 1 2|∇ρ|b 2− ∂ yρ∂b yρb      

where ρ is the dimensionless density, bb P is the dimensionless pressure and

b

vvv = (vbx,vby)

T the dimensionless velocity field. The Reynolds number, Re,

and the Weber number, W e can be expressed in terms of the dimensionless dynamic viscosity µ and the dimensionless capillarity coefficient bb λ as

Re = 1 b µ W e = 1 b λ 2.2. The FV-MLS method

2.2.1. Finite Volume formulation

In this work, we propose a discretization of the NSK equations described

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integral form of the system of conservation laws for each control volume I Z ΩI ∂uuu ∂t dΩ + Z ΓI F FFHHH− FFFEEE− FFFKKK · nnn dΓ = 000 (1) where ΩI is the control volume area, ΓI is the control volume boundary and

nnn = (nx, ny)T is the unitary exterior normal of the contour of ΩI. Moreover, FFFHHH,FFFEEE,FFFKKK are the convective, diffusive and Korteweg fluxes, respectively. Note that the integral over ΓI is performed over the interfaces (edges in 2D or faces in 3D) between two adjacent control volumes. In order to increase

the order of accuracy of a FV scheme there is a need to reconstruct the

variable uuu at the integration points on the interface. For convective terms,

this is usually achieved by using Taylor’s polynomials [15, 16, 17, 18, 19]. The reconstruction for diffusive and Korteweg terms will be discussed later.

We define a reference point (node), xxxI inside each cell (the cell centroid). The spatial representation of the solution is as follows: consider a function

uuu(xxx), given by its point values, uuuI = uuu(xxxI), at the cell centroids, with coordi-nates xxxI. The approximate function uuuh(xxx) belongs to the subspace spanned

by a set of basis functions {ΦI(xxx)} associated to the nodes, such that uuuh(xxx) is given by u uuh(xxx) = nxxx X j=1 Φj(xxx)uuuj (2)

which states that the approximation at a point xxx is computed using certain nxxx surrounding nodes. This set of nodes is referred to as the stencil associated

to the evaluation point xxx, and uuuj refers to the value of the function at the points of the stencil. The number of nodes of the stencil depends on the

order of the reconstruction, and will be discussed later.

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(1) we obtain Z ΩI ∂uuuh ∂t dΩ + Z ΓI F FFhHhHhH− FFFhEhEhE − FFFhKhKhK · nnn dΓ = 000 (3) where the superindex h indicates that the flux is computed with the

recon-structed function uuuh.

2.2.2. Moving Least Squares approximations

In equation (2) Φj(xxx) are the MLS shape functions. Here we only

in-troduce a brief description of the computation of MLS functions, and we refer the reader to [15, 16, 17, 18, 19] for a complete description of the

pro-cedure. To compute the MLS shape functions we define an m-dimensional basis, which in this case is defined as pppT(xxx) = (1, x, y, x2, y2, xy, ...) ∈ Rm. The dimension of the basis, m, determines the minimum number of points of the stencil. However, for stability reasons, this minimum number should

be increased. In this work we use a polynomial cubic basis, with a value of nxxx = 13 for convective fluxes and nxxx = 16 for the diffusive fluxes. Then, the

MLS-shape functions are defined as

Φ Φ

ΦT(xxx) = pppT(xxx)MMM−1PPP WWW (xxx)

where Pij = [pppT(xxxj)]i, is a m × nxxx matrix where the basis functions are evaluated at each point of the stencil, and MMM is the m × m moment matrix

given by

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The kernel function WWW determines the properties of the scheme, required

in the computation of ΦΦΦT(xxx). We have chosen to use an exponential kernel [19], defined as Wi(xxxj) = e−(sc) 2 − e−(dmc ) 2 1 − e−(dmc ) 2

with s = |xxxj− xxxi|, dm = max (|xxxj− xxxi|), with j = 1, . . . , nx∗, c = dm

2κ, and

κ is a shape parameter, which in this work is taken as κ = 3. Moreover, xi

refers to the position of the point where the MLS shape function is evaluated.

Using MLS reconstructions we obtain a continuous surface (volume in 3D) with the value of the reconstructed function. This surface/volume is

defined on a support given by the stencil of the approximation. Thus, from this surface/volume it is possible to define a single value of the reconstructed

function at integration points, and this is the procedure used here for diffusive and Korteweg fluxes. For the convective flux, we use a different procedure to

allow using the Riemann solver technology. For convective fluxes we “break” the MLS reconstruction to obtain the two Riemann states at the interface

by using Taylor series expansions, as explained in the following section.

2.2.3. Convective fluxes

For hyperbolic fluxes, we introduce a “broken” reconstruction in terms

of Taylor series, which approximates uuu(xxx) locally inside each cell I, and is discontinuous across cell interfaces. This procedure allows us to calculate

the numerical flux at each Gauss point at cell interfaces by using a Riemann solver. In general, we require the order of accuracy of the broken

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using Taylor series expansions; a quadratic reconstruction inside cell I, reads u u uhbI (xxx) = uuuhI + ∇∇u∇uuhI · (xxx − xxxI) + 1 2(xxx − xxxI) T HHHh (xxx − xxxI) (4)

where the gradient ∇∇∇uuuh

I and the Hessian matrix HHHh involve the successive derivatives of the continuous reconstruction uuuh(xxx), which are evaluated at the cell centroids using MLS.

For unsteady problems, additional terms must be introduced in equation (4) to enforce conservation of the mean

1 VI Z x x x∈ΩI uuu(xxx)dΩ = uuuI

where VI is the measure of the control volume cell.

Thus, the quadratic reconstruction for unsteady problems reads as

u uu(xxx) = uuuI+ ∇∇∇uuuI· (xxx − xxxI) + 12(xxx − xxxI)THHHI(xxx − xxxI)− − 1 2VI  Ixx ∂2uuu ∂x2 + 2Ixy ∂2uuu ∂x∂y + Iyy ∂2uuu ∂y2  with Ixx = R Ω(x − xI) 2dΩ, I yy = R Ω(y − yI) 2dΩ Ixy = R Ω(x − xI)(y − yI)dΩ

In this work, we have used the Rusanov Riemann solver [27] with the Li

and Gu’s fix for all speed flows [28], which can be written as

Θ Θ Θi+1 2 = 1 2(FFF hH++ FFFhH−) · nnn − 1 2S +∗ (uuu) (5) with S+ = max(|vvv+| + c+, |vvv| + c− ) (6)

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We define ∆∗(uuu) as follows ∆∗(uuu) =          b ρhb+ b ρhb− f (Ml) (ρbvbx) hb+− ( b ρvbx)hb−  f (Ml) (ρbvby) hb+− ( b ρvby)hb−           where f (Ml) =        min(Ml, 1), if ∂ bP ∂ρb > 0 0, if ∂ bP ∂ρb ≤ 0 (7) and Ml = |vbx|I+ |vby|I cI

Note that the Li and Gu’s fix is simply obtained by multiplying the mo-mentum difference term in the momo-mentum equations by the function f (Ml).

In equation (6) cI is the sound velocity on the cell I and |vvv| is the mod-ulus of the velocity vector at integration point and ∆(uuu) = (uuuhb+− uuuhb−). In interface regions, the sound velocity may become a complex number. In particular, this happens when ∂ b∂bPρ is negative. Thus, we define cI as follows

cI = v u u tmax ∂ bP ∂ρb, 0 ! (8)

Note that the use of Li and Gu’s fix allows to use this scheme in the interface region, where the equations are no longer hyperbolic. Then, introducing

equation (5) in equation (3) Z ΩI ∂uuuh ∂t dΩ + Z ΓI Θ ΘΘ(uuuhb+, uuuhb−) dΓ − Z ΓI FFFhEhEhE− FFFhKhKhK · nnn dΓ = 000

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Note that this dual continuous/discontinuous(for diffusive/Korteweg and

convective terms) reconstruction of the solution is crucial in order to

ob-tain accurate and efficient numerical schemes for mixed parabolic/hyperbolic

problems. The cell-wise broken reconstruction defined here is actually a piecewise continuous approximation to uuuh. As commented previously, the

advantage is that it allows to make use of Riemann solvers, and other stan-dard finite volume technologies, while keeping some consistency in terms of

functional representation.

2.2.4. Diffusive and Korteweg fluxes

For elliptic fluxes a continuous reconstruction is used with MLS directly

at cell interfaces. The numerical method proposed in this work, performs a centered, direct evaluation of the viscous fluxes at the quadrature points on

the edges using information from neighboring cells, through the use of MLS approximations. Thus, focusing on the Navier–Stokes-Korteweg equations,

the evaluation of the diffusive and Korteweg fluxes requires interpolating the derivatives of the dimensionless density and velocity vector bvvv = (vbx,vby)

T at each quadrature point xxxiq. Using MLS approximations, these values are readily computed as ∂xiρ =b nxxx X j=1 ∂xiΦj(xxxiq)ρbj (9) ∂xivbx = nxxx X j=1 ∂xiΦj(xxxiq)vbxj ∂xivby = nxxx X j=1 ∂xiΦj(xxxiq)vby j (10)

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function are computed at the quadrature point xxxiq. We refer the reader

to [15, 16, 17, 18, 19] for a complete description of the procedure of the computation of derivatives of MLS shape functions.

In all the MLS approximations of this work, we have used a cubic polyno-mial basis, which leads to third-order discretizations of the first derivatives

[29].

2.2.5. Pure Reconstruction test

In order to demonstrate the accuracy and the formal order of

reconstruc-tion of the variables and its derivatives, we perform a reconstrucreconstruc-tion test case. The 2D square domain Ω = [−1.0, 1.0] × [−1.0, 1.0] is discretized with

a set of structured meshes. We define the function to be reconstructed as

f (rrr) = a e−|rrr|22b (11)

where rrr is the radial distance rrr = p(x − 0.5)2+ (y − 0.5)2 , a = 1/(2bπ) and b = 0.20. First, we set the known value of the function to each cell

centroid. Next, MLS reconstruction and first, second and third derivatives of the function are computed at each integration point. Then, since the

analytical values are known the LN

2 for the reconstruction and derivatives errors for each mesh resolution is computed. Finally, once the errors are

known, the convergence rates are computed as

ON = log L N1 2 /L N2 2  log(pN2/N1) . (12)

where N1 and N2 are the number of control volumes for two different mesh resolution. LN

2 is define as the L2-norm of the error on the mesh with N

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function and its first, second and third derivatives, are plotted in Figure 1

for a cubic reconstruction. It is shown that the expected convergence rates are recovered. 256 1024 4096 16384 Mesh resolution 10-10 10-8 10-6 10-4 10-2 100 L2 error norm 1 2 3 4 f(x,y) xf(x,y) yf(x,y) 2 xxf(x,y) 2 yyf(x,y) 2 xyf(x,y) 3 xxxf(x,y) 3 yyyf(x,y) 3 xxyf(x,y)

Figure 1: 2D Reconstruction test cases using a cubic MLS reconstrucion. LN2 error norms

and convergence rates of the reconstructed function and its first, second and third deriva-tives.

3. A new Equation of State

In order to fully define the Navier-Stokes Korteweg system, it has to

be complemented with a non-monotone equation of state of pressure with density. Traditionally, the most commonly used is the van der Waals equation

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b P = 8 bTρb 27(1 −ρ)b −ρb 2 with b T = T Tcrit = 27R 8abT P =b 1 ab2P ρ =b 1 bρ

where R is the ideal gas constant and the parameters a and b are constants

that have different values for each substance.

Figure 2: Van der Waals isotherm corresponding to bT = 0.85. In the figure we show the

pure phases outside the saturation curve, vapor phase on the left and liquid phase on the right. Inside the spinodal curve, ∂ bP

b

ρ < 0 and therefore the EOS is of elliptic type. Between

both curves, the metastable regions take place.

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Under the critical temperature ( bT < 1), the pressure function for a van der

Waals fluid is non-monotone with density, unlike in a one-phase fluid. The function represented can be interpreted as a vapor phase in the range (0, ρbA)

and a liquid phase in the range (ρbB, 1). As it can be seen in figure 2, under the spinodal curve between ρbAand ρbB the curve is of elliptic type as

∂ bP ∂ρb

< 0.

In this region a decrease in the pressure of the substance corresponds to an increase in the density, which represents a non-physical behavior.

The saturation curve in figure 2 separates two different stable phases (pure phases), the vapor pure phase on the left and the liquid pure phase on the

right. The saturation curve represents the region where the thermodynamic equilibrium between both phases takes place, and thus they can coexist at the

same time. Inside the saturation curve, the van der Waals equation of state proposes a continuous and smooth transition between both phases, which

makes it adequate for diffuse interface modeling. The expression inside the saturation curve, though, does not represent a physical behavior, it is not

possible to obtain the real value in a laboratory and therefore there is not empirical data to compare with.

Nevertheless, van der Waals equation of state, although accurate for a great number of substances, does not reproduce precisely the specific case of

water. There are certainly other equations of state meant to adjust better to empirical data, such as the Modified Tait’s equation [30, 31], written in

equation (13), which represents the liquid phase of the water.

b P = K0 "  b ρ b ρl,sat(T ) N − 1 # + bPsat(T ) (13)

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for a certain temperature.

However, due to the absence of experimental data, more precise equations of state do not establish any expression for the substance under the saturation

curve. Hence, their use is not valid in a diffuse-interface model.

In figure 3, we present the isotherms at bT = 0.85 using the van der Waals

and Tait EOS compared with the experimental data obtained from the NIST database [32]. Figure 3 shows the inaccuracy problems that van der Waals

EOS present in pure phases outside the saturation curve, which are even greater as the temperature drops. Moreover, the van der Waals equation

states that the thermodynamic equilibrium between phases is produced at the saturation pressure bP = vdW bPsat, which does not match with the real

value bP = bPsat. For the temperature used in figure 3 ( bT = 0.85), the value of vdW bPsat has a relative error of 82.03%.

3.1. Objective and main ideas

In this work we propose a new algorithm to obtain more accurate equa-tions of state from data exclusively in the pure phases. In our approach,

the global EOS is defined by blending two different EOS that reproduce ac-curately both of the pure phases. Here, we specifically address the case of

water, and we use Tait’s EOS for the liquid phase and van der Waals EOS for the vapor phase. In order to be adequate for diffuse interface modeling,

the global equation of state must also describe the behavior of the fluid un-der the saturation curve. This is achieved by creating a reconstruction of

the equation as it is explained in the following paragraphs. A first approach to the problem addressed in this work is exposed in [33], where the author

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0 0.2 0.4 0.6 0.8 0

0.01 0.02 0.03

Figure 3: Comparison of the isotherms (at bT = 0.85) using the van der Waals and Tait

EOS for water, compared with the experimental data obtained from the NIST database.

With the aim of creating the global EOS, some requirements have to be verified, which can be listed as

1. In the global EOS, both the function and the first derivative must be

continuous (the EOS must be at least of differentiability class C1), as it is required for the discretization of the Korteweg tensor and to

ensure the continuity of the sound velocity. The reconstruction will be performed between the points ρbV and ρbL, both with the same pressure

b

Psat (figure 3) which are defined by the equations of state used in the pure phases.

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2. Additionally, the expression under the saturation curve must be defined

in such a way that the thermodynamic equilibrium is produced at the saturation points, at densityρbV for the vapor phase andρbLfor the liquid

phase, both with the same pressure bPsat (figure 2). In this work, this is achieved by fulfilling the Maxwell equal area rule [34], which states that

Gibbs free energy has the same value in two phases in equilibrium [35]. However, there are other possibilities, since there is some controversy

about the use of the Maxwell rule [34]. For the isothermal case, the Gibbs free energy reads as

G = P V − Z

P dV + ϕ(T )

where P = P (V ), which under the saturation curve can be modified

according to the reconstruction used.

Minimum values in Gibbs free energy function represent the

thermody-namic equilibrium of the substance [36]. In a one-component system with two phases at equilibrium, for a certain value of temperature and pressure

the Gibbs free energy for both phases must be equal. In figure 4 (a) we show for bT = 0.85 and bP = 0.187 a representation of the Gibbs free energy for the

van der Waals equation as a function of the volume bv = 1/ρ. For this combi-b nation of temperature and pressure, the Gibbs free energy function produces

two minima with the same value, representing two phases at equilibrium at

b

v = bvL and bv = bvV, which are the volume values of the saturation points.

Moreover, in figure 4 (b) we plot the van der Waals equation of state as function of the volume. From a geometrical point of view, the Maxwell rule

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b

Psat have the same area in the representation of pressure as function of the

volume, as it is shown in the two shaded areas in figure 4 (b).

(a) (b)

Figure 4: In (a), we show a representation of the dimensionless Gibbs free energy depending onbv with bT = 0.85 and bP = 0.187, which produces two minima with the same free energy value. The figure shows the state of a substance in which both liquid and vapor phases are stable and therefore the coexistence is produced. In (b), we plot the van der Waals equation of state depending on the volume, in which the two shaded areas delimited by

b

Psat= 0.187 are equal according to the Maxwell rule.

3. Generally, the reconstruction curve must keep the same shape of the

van der Waals equation of state, producing only two points with zero derivative inside the saturation curve, which are the spinodal points.

We will elude the creation of new points with zero derivative so as to avoid changes in the shape of the Gibbs free energy (see figure 4), which

is directly related to the mechanism of separation of the phases.

Note that the choice of the EOS under the saturation curve, between liquid and vapor phases, will inevitably affect the underlying physics of the

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substance within the interface, specifically, it will affect the predicted value

of the surface tension, which is related to the excess of the Gibbs free energy inside the interface. Previous works have addressed the modification of the

EOS under the saturation curve [33, 37]. In particular, in [37] the authors present a very exhaustive analysis of the thermodynamic effects of modifying

the equation of state under the saturation curve in the context of NSK equa-tions. How the expression under the saturation curve affects key variables in

the numerical method presented here will be discussed in Section 4.

3.2. Reconstruction function

In order to define the reconstruction function, we use a B-spline

approxi-mation due to the great control it provides over the final shape of the curve. In this work, B-splines are computed using the recursive Cox-de Boor

algo-rithm [38]. The blending curve S is defined as the sum of B-spline functions of order p (Ni,p(ξ)) multiplied by a vector-valued constants Ci, called control

points as S(ξ) = n X i=1 Ni,p(ξ)Ci

where n is the number of B-spline shape functions (NNN ) and points used in the interpolation.

To meet the continuity and derivability requirements of the EOS curve, the shape functions must be at least quadratic (p ≥ 2) so that Ni,p(ξ) ∈

Cp−1. To build the reconstruction we use n = 6 control points. Figure 5 shows schematically the kind of curve that we obtain for the EOS under the

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A

B

Figure 5: Sketch of the curve obtained for the EOS under the saturation curve using the

proposed reconstruction scheme. In the figure, Ci are the control points of the B-Spline

and δP and ∆P are parameters of the reconstruction. A and B are the spinodal points of the EOS, which are coincident with the maximum and minimum of the B-Spline curve.

The proposed procedure is summarized as follows

1. Control points C1 and C6 are placed at the saturation points, as they

are the beginning and the end of the reconstruction domain.

2. The position of the control points C2 and C5 determines the value of

the initial and final slopes of the curve respectively (S0(ρbV) = αV for vapor and S0(ρbL) = αL for the liquid phase) as it is seen in figure 5.

In order to achieve continuity of the first derivative in the global EOS, these points will be placed to match ∂ b∂bPρ derivatives at the saturation

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points of the reconstruction curve with those of the pure EOS.

3. The reconstruction curve S is tangent to the line segments C2C3, C3C4

and C4C5 in their midpoints (marked in figure 5). Since we set the points C2 & C3 and C4 & C5 to have the same ordinate value, two of

the tangent points have zero derivative, thus they are spinodal points (A and B in figure 5). The density value of the spinodal points (denoted

as ρbA and ρbB) is considered as a given data, and it is chosen so that the general shape of the global EOS is similar to the van der Waals

EOS, as it is our reference. Therefore, we keep the same proportional relation between ρbV, ρbA, ρbB and ρbL as in vdW EOS.

To completely define the position of the spinodal points, we establish as variables the differences of pressure between the points C1 and C2, which we

call δP , and the difference of pressure between spinodal points, which we call ∆P , as we show in figure 5. At this point, for a certain value of δP and ∆P

we have already established the points C2 and C5, as their pressure is known ( bPsat+ δP for C2 and bPsat+ δP − ∆P for C4) and they belong to the tangent

lines of the EOS at the saturation points.

4. Finally, C3and C4are chosen such as the midpoints of the line segments

C2C3 and C4C5 coincide with the spinodal points. This condition can be expressed as C2A = AC3 and C4B = BC5.

Thus, in brief, the pair    b ρ b P   

for each control point is given by

C1 =    b ρV b Psat    C2 =    b ρV +αδP V b Psat+ δP    C3 =    2ρbA− (ρbV + δP αV) b Psat+ δP   

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C4 =    2ρbB− (ρbL− ∆P −δP αL ) b Psat+ δP − ∆P    C5 =    b ρL− ∆P −δPα L b Psat+ δP − ∆P    C6 =    b ρL b Psat    (14)

The position of the control points depends on two parameters, ∆P and

δP . In order to completely define the global EOS we keep ∆P as a free parameter, which allow us to have control on the final shape of the

recon-struction. The choice of this parameter will affect important thermodynamic properties of the substance, such as the surface tension or the width of the

interface.

The value of δP is computed to fulfill the Maxwell’s equal area rule, which

is expressed in equation (15).

Z bvV

b vL

( bP (bv) − bPsat)dbv = 0 (15)

In equation (15), bP (bv) is obtained through the equation of state, which

is dependent on the volume v = 1/b ρ. Moreover,b bvV and bvL are the values of the volume at saturation points.

Expanding equation (15) with the values of the control points in the previous expressions (14), results in an analytic expression of δP depending

on ∆P , and therefore the curve is completely defined. How ∆P affects the

stability and numerical solution of the model will be analyzed in Section 4. A major advantage in using this algorithm is the great control it provides

on the final curve and the simplicity of its application. B-Splines allow us to easily define the initial and final slopes necessary to meet the smoothness

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reconstruction procedure allows us to use different equations of state for

each substance and make them suitable for diffuse interface modeling. An additional advantage of the proposed approach is that the reconstruction

curve is completely defined by an analytic algorithm.

In figures 6-9 two examples of the equation of state obtained using this

algorithm are shown. In the first example, the temperature is set at bT = 0.85 and ∆P = 0.010. Moreover, in figure 7 the same EOS as a function of the

volume (instead of the density) is plotted. In this last figure, it can be seen that the Maxwell equal area rule is fulfilled.

In the second example, shown in figure 8, the equation of state obtained for the temperature bT = 0.65 and ∆P = 0.010 is drawn. The same EOS

as a function of the volume is also shown in figure 9. In this case we use a logarithm scale for the representation of the volume due to the large values

of this variable.

4. Numerical results

In this section, several numerical examples are presented to show the

ability of the proposed method to simulate two-phase flow phenomena. The numerical examples are computed on a square-shaped computational domain

Ω = [0, 1] × [0, 1], in which we impose periodic boundary conditions in all directions. For the examples the physical parameters Ca, W e and Re are

defined as follows Ca = h 2 L2 0 W e = 1 Ca Re = 2 √ W e L0 = 1 (16)

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Figure 6: Equation of state for a temperature of bT = 0.85. In the equation, the vapor branch is modeled with the van der Waals equation of state, and for the liquid branch we use the Modified Tait’s equation. Under the saturation curve, the algorithm presented in this work is used with ∆P = 0.01.

where L0 is an arbitrary length. The numerical method includes a

dimen-sionless characteristic length h to scale with the capillarity number, which is directly related to the width of the interface. One of the challenges that

diffuse-interface models face is that, in order to be realistic, the interface must be extremely small, which would need very fine meshes to be captured

[37]. In this work, following [14], the capillarity number is related with the computational mesh as

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0 5 10 15 20 25 -0.005 0 0.005 0.01 0.015 0.02 0.025

Figure 7: Equation of state for a temperature of bT = 0.85 depending on the volume. The

vapor branch of the EOS is modeled with the van der Waals equation of state, and for the liquid branch the modified Tait’s equation is used. Under the saturation curve, the algorithm presented in this work is used with ∆P = 0.01. The figure shows that using the proposed algorithm, the Maxwell’s equal area rule is fulfilled.

where Ai is the area of the i -th element of the computational mesh. The

constant parameter C, which relates the characteristic length to the mesh, takes different values for different authors [14, 39]. Here, we set C = 1.

Hence, in the numerical simulation of a diffuse-interface model like this, the characteristic length is a very important attribute to simulate successfully

the diffuse interface. As it is shown in [39, 26], the condition that has to be accomplished is that

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Figure 8: Equation of state obtained with the algorithm we present in this work for a

temperature of bT = 0.65. The vapor branch of the EOS is modeled with the van der

Waals equation of state, and for the liquid branch the modified Tait’s equation is used. Under the saturation curve, the algorithm presented in this work is used with ∆P = 0.01.

h ≤ %b W e

This impose an upper bound to the characteristic length, where % is ab dimensionless positive constant, which usually takes the value of % = 1. Notb

accomplishing this condition may produce non-physical oscillations that do not simulate accurately the different states.

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100 101 102 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02

Figure 9: Equation of state for a temperature of bT = 0.65 as function of the volume. The vapor branch of the EOS is modeled with the van der Waals equation of state, and for the liquid branch the modified Tait’s equation is used. Under the saturation curve, the algorithm presented in this work is used with ∆P = 0.01. In this figure, the volume is represented in a logarithm scale due to the large values of this variable.

4.1. Manufactured Solution

The first numerical example is the manufactured solution test case. The aim of this numerical test is to verify the accuracy of the proposed scheme in the treatment of the convective, diffusive and Korteweg-like terms. The equation to be solved is ∂ρ ∂t + a ∂ρ ∂x − b ∂2ρ ∂x2 + c ∂3ρ ∂x3 = S (18)

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where we consider the manufactured solution

ρ (x, t) = 1

2sin (2x) e −t

(19)

The source term S is computed by substituting the manufactured solution

(19) in equation (18), to obtain

S(x, t) = −1 2e

−t

(sin(2x) − 2a cos(2x) + 8c cos(2x) − 4b sin(2x))

First, we begin with the one-dimensional test case. The computational do-main is Ω = [0, π]. Boundary conditions are prescribed using the analytical

solution.. We begin with the linear convection problem, where the solution

is computed with the parameters a = 1, b = 0, c = 0. For the pure diffu-sion problem we set a = 0, b = 0.1, c = 0 and with the Korteweg-like term

a = 0, b = 0, c = 0.01. All the solutions are computed until t = 0.1. As the analytical solution is known, we can evaluate the order of accuracy of the

scheme. The obtained L2 norms of the error in ρ are shown in Table 1.

Convection Diffusion Korteweg-like term

M esh L2-norm Order L2-norm Order L2-norm Order

16 1.62E-04 – 1.32E-04 – 1.58E-04 –

32 1.88E-05 3.10 3.96E-05 1.74 4.04E-05 1.97

64 2.25E-06 3.07 1.04E-05 1.93 1.03E-05 1.97

128 2.73E-07 3.04 2.63E-06 1.98 2.63E-06 1.98

256 3.37E-08 3.02 6.60E-07 1.99 6.64E-07 1.98

Table 1: 1-D Manufactured solution. Accuracy order and L2-norm of ρ for convection, diffusion and Korteweg-like behaviour.

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Next, we analyze the order of convergence of the system when convective, diffusive and Korteweg-like terms take place at the same time. Two set of parameters, a = 1, b = 0.1, c = 0.01 and a = 1, b = 1, c = 1 are chosen. The

obtained L2 norms of the error in ρ are shown in Table 2.

a = 1, b = 0.1, c = 0.01 a = 1, b = 1, c = 1

M esh L2-norm Order L2-norm Order

16 3.42E-04 – 1.41E-02 –

32 6.97E-05 2.29 3.80E-03 1.89

64 1.59E-05 2.13 9.78E-04 1.96

128 3.83E-06 2.06 2.48E-04 1.98

256 9.38E-07 2.03 6.23E-05 1.99

Table 2: 1-D Manufactured solution. Accuracy order and L2-norm of ρ for convection,

diffusion and Korteweg-like behaviour.

In Tables 1 and 2 it is shown that the expected convergence rates are recovered.

In order to analyze the accuracy of our scheme on a multi-dimensional setting, we have extended this test case to 2D. The computational domain Ω = [0, π]×[0, π] is discretized with a set of structured meshes. At the bound-aries, we prescribe the analytical solution. Accuracy order and L2-norm of the error are shown in Table 3, where the expected rates are recovered.

4.2. Single bubble equilibrium

In the second example, we impose an initial condition which is not a

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a = 1, b = 0.1, c = 0.01 a = 1, b = 1, c = 1

M esh L2-norm Order L2-norm Order

16 × 16 4.72E-04 – 2.53E-02 –

32 × 32 7.29E-05 2.69 5.62E-03 2.17

64 × 64 1.72E-05 2.08 1.35E-03 2.06

128 × 128 4.28E-06 2.01 3.40E-04 2.02

Table 3: 2-D Manufactured solution. Accuracy order and L2-norm of ρ for convection,

diffusion and Korteweg-like behaviour.

In this simulation we show the evolution of a two-dimensional vapor bubble centered in the computational domain with a fixed temperature bT = 0.85.

For the initial conditions, we set the radius of the bubble R = 0.284. The interface is shaped in the initial condition using a hyperbolic tangent profile,

which is a good approximation although it does not fulfill the equation sys-tem. Additionally, the initial velocity field is set to be null. These conditions

are written as follows

b ρ(xxx, t = 0) = 0.4148 + 0.3677  tanh d(xxx) − R 2h  (vx, vy)T = (0, 0)T

where d(xxx) is the Euclidean distance between xxx and the center of the

computational domain (0.5, 0.5). The computational mesh is composed by 642 elements.

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3, with the Tait equation of state for the liquid phase, the van der Waals

EOS for the vapor phase and using the reconstruction with ∆P = 0.01. The result of the simulation is shown in figure 10, where we represent the steady

state of the density in the two-dimensional domain. The interface separating both liquid and vapor phases is clearly seen in the figure. We also note the

perfectly circular shape of the bubble, as expected.

Figure 10: steady state of a single bubble immersed in liquid: Density field. For this

simulation we use 642 elements and ∆t = 10−4. The EOS employed is the one plotted in

figure 6.

It is important to remark that, using the algorithm presented here, for a

given temperature, we are able to obtain different EOS for each value of the parameter ∆P . However, as it is pointed out in [37], changing the expression

under the saturation curve will inevitably modify the underlying physics of the interface. This is addressed in the following sections.

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4.2.1. Influence of ∆P

The equation which describes the mechanics of a vapor bubble immersed

in a liquid is the Rayleight-Plesset equation [40, 41], which in the steady state is simplified to the Young-Laplace equation [42, 43].

| PV − PL|= (n − 1)σk (20)

In equation (20), | PV − PL | denotes the difference of pressure between two phases in equilibrium, which in this example are vapor and liquid. The parameter n is the number of dimensions in the problem, k = 1

R is the mean curvature of the interface and the constant σ is the surface tension. In consequence, we can rewrite the expression for this case as

| PV − PL | R = σ (21)

This relation implies that for a certain temperature, we can find a value for the surface tension σ measuring the radius of the bubble in the steady

state of the numerical simulations, in order to find a relation between the surface tension and the value of ∆P in our EOS algorithm. In figure 11

we plot the EOS using different values of the parameter ∆P . Note that, as all the computations are performed for the same temperature bT = 0.85, the

EOS for the pure phases remains unaltered, only changing the shape of the EOS under the saturation curve.

In figure 12, the steady state of the single bubble equilibrium test com-puted with the different EOS plotted in figure 11 is shown. A cut at y = 0.5

of the different density profiles obtained is displayed in figure 13. As it is seen in the figures, both the width of the interface and the radius of the bubble

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 -0.035 -0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015

Figure 11: Representation of different EOS obtained using the algorithm proposed for the

same value of temperature bT = 0.85 with different values of the reconstruction parameter

∆P .

change as ∆P varies.

In order to get the radius of a bubble in a diffuse interface model, it is necessary to set a certain criterion to establish the boundary of the phases,

as due to the nature of the diffuse interface models this separation remains unclear. As it is presented in [26], we define two points ˜x1 and ˜x2, which will

mark the separation between the phases, as ρ(˜bx1) = ρ(˜bx2) = (ρbL −ρbV)/2 where 0 ≤ ˜x1 ≤ 0.5 ≤ ˜x2 ≤ 1. Therefore, the boundary between the phases is located in the midpoint of the density transition. Finally, we define an approximate value of the radius as

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(a) ∆P = 0.01 (b) ∆P = 0.02

(c) ∆P = 0.03 (d) ∆P = 0.04 (e) ∆P = 0.05

Figure 12: steady state of a single bubble immersed in liquid: Density field for different

values of ∆P in the reconstruction of the EOS and a fixed temperature of bT = 0.85 using

the EOS plotted in figure 11. A computational mesh with 642elements is used for all the

computations.

˜

R = | ˜x1− ˜x2 | 2

Because of the nature of a numerical simulation, we are only able to obtain an approximated value of the radius ˜R, and thus an approximated

value of the surface tension ˜σ, computed with the relation shown in equation (21). The values of these parameters are listed in table 4.

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Figure 13: steady state of a single bubble immersed in liquid: Cut of the Density at y = 0.5 for different values of the parameter ∆P . In the examples we use the EOS plotted in figure

11. We set the same initial conditions, temperature ( bT = 0.85) and a mesh composed by

642 elements.

the interface W . As in the case of the radius, it is necessary to establish a

criterion to get this parameter from the numerical results. We set another two different points,bx1 andxb2, such asρ(˜bx1) = ρbL−0.05(ρbL−ρbV) andρ(˜bx2) = b

ρV + 0.05(ρbL−ρbV), with 0 ≤ ˜x1 ≤ ˜x2 ≤ 0.5 . By choosing these points, we ensure that the 90% of the variation is produced within the interface. Hence,

we define an approximate value of the interface width as

˜

W =| ˜x1 − ˜x2 |

The variation of ˜W depending on ∆P is also listed in table 4.

In table 4 it is shown that the approximated surface tension ˜σ increases

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b T ∆P R˜ W˜ | bPV − bPL| σ˜ 0.85 0.05 0.2823 8.537 × 10−2 5.534 × 10−3 1.562 × 10−3 0.04 0.2817 9.747 × 10−2 4.908 × 10−3 1.383 × 10−3 0.03 0.2801 11.30 × 10−2 4.234 × 10−3 1.186 × 10−3 0.02 0.2781 13.59 × 10−2 3.491 × 10−3 0.9707 × 10−3 0.01 0.2722 19.12 × 10−2 2.481 × 10−3 0.6752 × 10−3

Table 4: Approximated data from the numerical examples of a single bubble in equilibrium.

The numerical simulations are computed with the same temperature bT = 0.85 and a time

step of ∆t = 10−4. In this table we show the relation between the radius of the bubble

˜

R, the width of the interface ˜W and the surface tension ˜σ with the parameter of the

reconstruction proposed ∆P .

relation between the surface tension and the parameter ∆P , for bT = 0.85,

can be accurately approximated with a square root function, as

˜

σ = 0.0069√∆P (22)

as it is shown in figure 14.

Conceptually, higher values of the surface tension can hold bubbles with

larger radii. In the limit with no surface tension the bubble is not stable under the pressure the liquid imposes.

In table 4 it is also shown that the parameter ∆P has also an influence on the width of the interface. This fact was already addressed in [37], where

the author proposes a method to enlarge the interface by reducing drastically the difference of pressure between the spinodal points. From the data shown

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0 0.01 0.02 0.03 0.04 0.05 0.06 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 10 -3

Figure 14: Relation between the reconstruction parameter ∆P and the approximated

value of the surface tension ˜σ for bT = 0.85 and a mesh composed by 642 elements. The

straight line is the curve obtained using equation (22) and the red diamonds are the results obtained from the numerical simulations.

and ∆P , for bT = 0.85, can be approximated as

˜

W = 0.0193/√∆P (23)

as it is shown in figure 15.

Note that relations (22) and (23) are obtained empirically for a very specific case, and thus this relations may change for other problems.

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0 0.01 0.02 0.03 0.04 0.05 0.06 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Figure 15: Relation between the reconstruction parameter ∆P and the approximated

value of the interface thickness ˜W for bT = 0.85 and a mesh composed by 642 elements.

The straight line is the curve obtained using equation (23) and the blue diamonds are the results obtained from the numerical simulations.

4.2.2. Influence of the characteristic length

For the numerical method to be precise, it needs a thin and therefore

realistic representation of the interface [37]. On the other hand, if the mesh is not fine enough to capture the interface, non-wanted oscillations may appear

in the solution and affect its accuracy. Although we have seen that the parameter ∆P has an influence on the width of the interface, the method

includes a characteristic length h (see equations (16) and (17)) to adapt the interface to the mesh. However, the characteristic length will also affect the

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In order to study the relation between the surface tension and h, we

have performed several simulations with the same temperature and the same ∆P . The objective of this study is to find a combined equation which

re-lates the surface tension to both h and ∆P for the isothermal version of the method. With simulations for meshes composed by 322, 482, 642, 962 and

1282 elements, related with h through (17), for bT = 0.85 we can accurately approximate the surface tension as

˜ σ = 0.4442h√∆P (24) as it is shown in figure 16. 0 0.06 1 2 10-3 3 0.04 4 0.02 0.035 0.03 h 0.025 0.02 0.015 0 0 0.005 0.01

Figure 16: Relation between surface tension and the reconstruction parameter ∆P and the characteristic length of the mesh h. In the figure it is seen that the surface tension increases linearly with h and has a relation with ∆P of a square root function.

Figure 16 shows empirically that the surface tension increases linearly

with the characteristic length and has the same relation with ∆P that we have shown in figure 14. These relations are in agreement with the results

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tension, we find the relation of ∆P and h with the thickness of the interface ˜

W . We find that

˜

W = 1.278h/√∆P (25)

This relation is shown in figure 17.

0 0.5 0.03 1 1.5 0.02 0 h 2 0.01 0.02 0.01 0.03 0.04 0.05 0 0.06

Figure 17: Relation between the thickness of the interface and the reconstruction parame-ter ∆P and the characparame-teristic length of the mesh h. In the figure it is seen that the surface tension increases linearly with h and has a relation with ∆P of an inverse square root function.

Note that in a numerical simulation, one of the most important factors is the size of the mesh. For the practical application of the methodology

presented in this work, we propose to select a suitable mesh and then select the reconstruction parameter ∆P in order to obtain the desired value of the

surface tension using equation (24). Once h and ∆P are selected, the width of the interface is established automatically. It is important to remark that

this have some consequences on the underlying physics that the method is able to represent, as presented in [37].

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specific case, and thus this relations may change for other problems.

4.2.3. Effects of temperature

According to equations (24) and (25), low values of the characteristic length h or great values of ∆P lead to thinner interfaces and therefore to

the need of finer meshes. For example, in figure 13 it is seen that for the highest value of ∆P (∆P = 0.05) a slight oscillations occurs. This is a

well-known problem with numerical simulations using the van der Waals equation of state, where the difference of pressure between the spinodal points creates

very thin interfaces difficult to capture by the mesh, a problem which grows exponentially when the temperature decreases. With the method proposed

in this work, we are able to control the expression of the EOS under the saturation curve, and therefore we can partially solve this problem, being

able to perform numerical simulations with lower temperatures. In figure 8, it is seen that for a lower value of the temperature ( bT = 0.65), the difference

of pressure of the spinodal points increases exponentially in the van der Waals EOS, whereas using the proposed method we are able to control this

attribute.

To analyze the effects of temperature variations, we show in figure 18 the

density profile of the numerical simulation of the single bubble equilibrium for different values of the temperature and the same ∆P = 0.01 with the

same computational mesh with 642. As in the previous subsections, different values of the parameters are collected in table 5. It is appreciated that,

although the interface width grows when the temperature drops, decreasing the value of the temperature may produce non-wanted oscillations. In figure

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density appear.

(a)

(b)

Figure 18: Single bubble problem: In (a), we show we show a cut of the density at y = 0.5 for different values of the temperature. For this examples we use the same value of the

reconstruction parameter ∆P and the time step ∆t = 10−5. In (b), we show a zoom of

the rectangle marked in (a).

This problem is usually soften by refining the grid, searching for a more

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elimi-b T ∆P R˜ W˜ | bPV − bPL| σ˜ 0.85 0.01 0.2725 0.1888 2.996 × 10−3 0.8164 × 10−3 0.80 0.01 0.2767 0.2003 3.104 × 10−3 8.590 × 10−3 0.75 0.01 0.2737 0.2148 3.639 × 10−3 9.962 × 10−3 0.70 0.01 0.2706 0.2198 4.139 × 10−3 1.120 × 10−3 0.65 0.01 0.2675 0.2255 4.622 × 10−3 1.236 × 10−3

Table 5: Approximated data obtained from the numerical simulation of a single bubble immersed in a liquid. In this table we represent the relation between the radius of the bubble ˜R, the width of the interface ˜W and the surface tension ˜σ with the temperature

b

T . For the simulation, the parameter of the interpolation ∆P is remained constant and the time step employed is ∆t = 10−5.

nate to a great extent the dependence on the mesh by adjusting the value of the parameter ∆P . By reducing the value of ∆P we are able to achieve

nu-merical solutions at lower temperatures, taking into account that decreasing ∆P makes the interface become wider and also reduces the surface tension

σ.

In figure 19 we show a numerical simulation of the single bubble

equilib-rium with different sizes of the mesh (and thus different values of h, equation (17)) and different values of the parameter ∆P at bT = 0.80. For a

computa-tional mesh composed by 642 elements and ∆P = 0.03, spurious oscillations are produced. As shown in the figure, making ∆P smaller can reduce the

oscillations to the same magnitude as reducing the value of h using more elements in the mesh.

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(a)

(b)

Figure 19: Single bubble problem: In (a), we show a cut of the density at y = 0.5

for a temperature bT = 0.80, using a mesh composed by 642 and 1282 elements and the

EOS computed with ∆P = 0.03 and ∆P = 0.01. In (b) we show a zoomed view of the rectangle marked in (a). Spurious oscillations that appear in the simulation with a mesh

of 642 elements and ∆P = 0.03 can be soften not only using a thinner mesh, like the one

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4.3. Coalescence of two bubbles

In the previous examples, a solitary bubble grew or shrank until it ar-rived to the steady state. With two or more bubbles sharing the same space,

the smaller bubbles will collapse due to the great pressure larger bubbles impose. When the vapor bubbles are close enough (a distance in the same

order of magnitude of the thickness of the interface), they will merge to form a lonely and larger bubble which will evolve to a circular shape [14]. To

sim-ulate this phenomenon of coalescence, we will place two vapor bubbles in the computational domain Ω. Their centers are placed at O1 = (0.375, 0.5) and

O2 = (0.775, 0.5). Their radii are R1 = 0.25 and R2 = 0.10 respectively. The initial condition is establish as in the previous examples using a hyperbolic

tangent profile to model the interfaces, and it reads as

b ρ(xxx, t = 0) = 0.04718 + 0.3676  tanh d1(xxx) − R1 2h  + tanh d2(xxx) − R2 2h  (vx, vy)T = (0, 0)T

where di(xxx) is the Euclidean distance between xxx and Oi, with i = 1, 2. The computational domain is discretized using both a 1282 and a 2562 grid.

In figures 20 and 21 the evolution of the density and pressure fields in time are shown using a computational mesh composed by 2562elements. These figures

show the ability of the numerical method to reproduce abrupt variations in the topology of the problem. The results obtained using the proposed

methodology are in agreement with those of [14].

According to the Young-Laplace equation (20), the difference in pressure

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(a) t=0 (b) t=1.50

(c) t=3.75 (d) Steady state

Figure 20: Coalescence of two bubbles: Evolution of the density field. Results on a 2562

grid.

the bubble. As σ is a constant parameter only depending on the equation of state and the capillarity number, we can establish a relation with the

approximated surface tension ˜σ calculated in the coalescence example and the single bubble equilibrium as

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(a) t=0 (b) t=1.50

(c) t=3.75 (d) Steady state

Figure 21: Coalescence of two bubbles: Evolution of the pressure field. Results on a 2562

grid.

where the sub-index 1 and 2 refer to two different examples with the same EOS. In order to find the error of the numerical solution, we check if

the results verify the Young-Laplace equation by simulating the coalescence of the bubbles with the same conditions that the single bubble equilibrium.

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| bPV − bPL|1 ' 1.220 × 10−3 R˜1 ' 0.2805 | bPV − bPL|2 ' 1.287 × 10−3 R˜2 ' 0.2684

which implies that, for this conditions, the Young-Laplace equation is

verified with a relative error of 9.62 × 10−3.

Conclusions

In this work we have proposed a new high-order accurate finite volume formulation for the resolution of Navier-Stokes Korteweg equations, based

on the Moving Least Squares (MLS) approximations. The Moving Least Squares approach is employed in order to obtain high-order approximations

and derivatives which allow us to obtain a high-order numerical method in both structured and non-structured meshes.

Moreover, we have presented a new algorithm to obtain more precise equations of state. In this method, we use more accurate equations of state

in pure phases joined together using a B-Spline reconstruction, creating new expressions of the equation of state under the saturation curve. The main

advantage of this algorithm is the ease of its application and his versatility, since to be used is only necessary the saturation points and the slopes of the

equation of states. Besides, although in this work we apply this algorithm to the specific case of water, it can be used for a great range of substances as

well.

Finally, we have shown some numerical examples of the formulation,

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empirically a relation between the parameters used in the interpolation and

important variables in the numerical calculus, such as the radius of the bub-ble, the width of the interface and the surface tension. Having the control

of the expression under the saturation curve allows us to select previously desirable values for this variables. Furthermore, using the algorithm

pro-posed also allows us to soften the dependence on the mesh of the numerical simulations.

Acknowledgments

This work has been partially supported by the Ministerio de Econom´ıa y Competitividad (grant #DPI2015-68431-R) and#RTI2018-093366-B-I00 of

the Ministerio de Ciencia, Innovaci´on y Universidades of the Spanish Gov-ernment and the Universidade da Coru˜na and by the Conseller´ıa de

Edu-caci´on e Ordenaci´on Universitaria of the Xunta de Galicia (grants # GRC2014/039 and # ED431C 2018/41), cofinanced with FEDER funds and the

Universi-dade da Coru˜na. Xes´us Nogueira also acknowledges the funding provided by the Xunta de Galicia through the program Axudas para a mellora, creaci´on,

reco˜necemento e estruturaci´on de agrupaci´ons estrat´exicas do Sistema uni-versitario de Galicia (reference # ED431E 2018/11).

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Figure

Figure 1: 2D Reconstruction test cases using a cubic MLS reconstrucion. L N 2 error norms and convergence rates of the reconstructed function and its first, second and third  deriva-tives.
Figure 2: Van der Waals isotherm corresponding to T b = 0.85. In the figure we show the pure phases outside the saturation curve, vapor phase on the left and liquid phase on the right
Figure 3: Comparison of the isotherms (at T b = 0.85) using the van der Waals and Tait EOS for water, compared with the experimental data obtained from the NIST database.
Figure 4: In (a), we show a representation of the dimensionless Gibbs free energy depending on bv with T b = 0.85 and Pb = 0.187, which produces two minima with the same free energy value
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